Analysis of couplers composed of a fibre half-block with a slab overlay: effects of curvature of the fibre and asymmetry of the slab waveguide

S. Zheng G.P. Simon L.N. Binh

Indexing terms: Coupled-mode theory, Fibre optics, Light coupling, Slab waveguide

Abstract: Light coupling between a single-mode fibre half-block and a single-mode asymmetric slab waveguide is analysed theoretically using coupled-mode theory. In comparison with pre- vious work where the fibre is straight or the slab waveguide is symmetric, the characteristics of coupling are distributed along the direction of light propagation and are strongly dependent not only on the relative values of the refractive indices of the fibre core and the slab, but also on the asymmetry of the slab waveguide, that is, the relative values of the refractive indices of the clad- dings on both sides of the slab.

1 Introduction

Much interest, both experimental and theoretical, has recently been shown in the coupling of light in composite fibre-slab waveguide structures. These basically consist of a single-mode fibre and a dielectric overlay, which leads to a variety of potential applications [l-91. A theoretical investigation on light coupling and propagation in such composite fibre-slab waveguide couplers has been pre- sented by Marcuse [6] for the case where the slab wave- guide is symmetric, namely, the refractive indices on both sides of the slab are the same. Recent experimental results [7, 81 of distributed coupling between a curved single- mode fibre and a symmetric slab waveguide have con- firmed the validity of the Marcuse's model by showing very good agreement between the two, demonstrating the relationship between the light power remaining in the fibre and the refractive index of the slab. To be able to account for some practical situations where the slab Naveguide is not symmetric, we have reported a modifi- cation of the coupled-mode and compound-mode the- ories so that they become applicable to the composite fibre-slab couplers in which the refractive indices on both sides of the slab are different [SI. New, closed-form ana- lytical expressions of coupling coefficients were obtained and a parameter was specifically introduced to quantify

0 IEE, 1995 Paper 20385 (E13), first received 4th July 1994 and in revised form 24th April 1995 S. Zheng and G.P. Simon are with the Department of Materials Engin- eering, Monash University, Clayton, Victoria 3168, Australia L.N. Binh is with the Department of Electrical and Computer Systems Engineering, Monash University, Clayton, Victoria 3168, Australia

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the asymmetry of the slab overlay. The strong depend- ence of the field and power distributions on the asym- metry of the slab waveguide have been observed by solving the coupled-mode and compound-mode equa- tions numerically.

In this paper the distributed coupling in composite waveguides composed of a single-mode fibre half-block with a single-mode asymmetric slab overlay is investig- ated theoretically. The coupled-mode theory is applied by using generally an asymmetric refractive-index profile of the slab waveguide and our closed-form expressions of the coupling coefficients [9]. The dependence of the dis- tributed light coupling, power distribution and transfer on the asymmetry of the slab overlay is illustrated and compared with the special case where a straight fibre is assumed in the fibre-slab waveguide coupler [6,9].

2 Theory

Figs. la and b are schematics of the fibre coupler half- block with a dielectric slab overlay. If the single-mode fibre and the slab waveguide are far apart from each other, it is said that they are unperturbed or uncoupled. In the coordinate system shown in Fig. 1, the dominant LP,, mode (the transverse electric field) of the isolated

I c "0

1 t ^ . I

"c

" 0

X Z J t &

Y

a

X t

I I 1 I I v

n. I J. I 1

c "f Z

b

Fig. 1 waveguide parameters a Cross-sectional view b Longitudinal view

Geometry of the fibre-slab waveguide coupler and the optical

IEE Proc.-Optoelectron., Vol. 142, No. 4, August 1995

fibre is given by

where J o and J, are the Bessel functions of the first kind, KO is the modified Bessel functions of the second kind and N I = y, J,(k,a)/J(rr)V, J,(k,a) is the normalisation contant of the fibre mode. The parameter a denotes the fibre radius, k; = n; k2 - @, and y; = p;, - nf k2 are constants in which k = 2n/A (A is the free-space light wavelength), p,, is the propagation constant obtained from the LPol mode dispersion equation [lo] and n, and n, are the refractive indices of the fibre core and cladding, respectively. V, = k d ( n ; - nf) is a dimensionless wave- guide parameter related to k, and y, via

V ; = a2(k; + y;) ( W I In the same coordinate svstem. the guided mode of the

with 2 0 representing the distance between the two per- fectly conducting planes hypothetically imposed so that S , = 0 at y = +D. Therefore, the geometric (vectorial) relationship between ps, {p,} (0, 1, 2, ...,) and an = (n + 1/2)n/D, (n = 0, 1, 2, ...) can be illustrated by Fig. 2, that is. the following relation between p, and {p,} (n = 0,

Psr

slab waveguide with a transverse or&r [SI n is obtained in the form

letric relations between the propagation constant B, ofthe ‘le axial component (0,)

for x < a + s

S,(x, y) = N , cos (any) {k , cos [k,(x - a - s - t ) ] - y o sin [k,(x - a - s - t)]} for a + s < x < a + s + t (2a)

for x > a + s + t

+ yr Y ~ C ) D ] ” ~ is the normalisation contant. The param- eters t denotes the slab thickness, s the minimum distance

where n = 0, 1, 2, ... and N , = [2y,y,/(y, + yc

between the surface of the fibre core and that of the polished flat, k,‘ = nfk2 - pi , y.f = p,’ - nfk2, y: = pf - n:k2 are constants in which p, IS the propagation con-

stant of the slab mode. n. and no are the refractive indices of the two waveguides will overlau and interact. That is, , -

of the slab and the overlay claddin respectively. V, = k t d m ) / 2 and V, = kt,/&)/2 are dimension- less waveguide parameters related to k, , yc and yo via

t2 = 7 (kf + yf)

The propagation contant of the slab mode of the mth order 8, can be obtained from the well known dispersion equation of the asymmetric slab waveguide

(m = 0, 1,2, . . .) ( 2 4 Note that only those slab modes with even symmetry along the y direction are adopted in expr. 2a because of the even symmetry of the LP,, mode [6] and the slab modes with propagation constant p, is free to travel in all possible directions in the plane of the slab, namely in the (y, z) plane but at certain angles to the z axis (Fig. 1) and thus forms a continuum of guided-modes. To obtain the discrete guided-modes {S,} (n = 0, 1, 2, . . .) in expr. 2a, characterised accordingly by {pan} (n = 0, 1, 2, . . .), a set of longitudinal propagation constants as projections of p, on the z axis, a parameter D is introduced as in [6-91

I E E Proc.-Optoefectron., Vol. 142, No . 4 , August 1995

the two wavegiides are now perturbed or coupled. Unlike the conventional single mode fibrefibre direc- tional couplers, fibreslab waveguide coupler involves coupling between infinitely many modes [SI.

In scalar approximation, the total transverse electric field of the fibre-slab structure can be represented by a superposition of the fibre and slab modes

where ao(z), {b,(z)} are the z-dependent mode amplitudes and the following coupled mode equations can be derived C6,9l.

bm = - -i 1 ( 6 m b s n + QsAbm - i K s m 0 a, ( 4 4

where a6 = da,/dz, b, = dbddz, pro and /?., (m = 0, 1, 2, . . .) are the propagation contants of the dominant LP,, mode of the optical fibre and the slab mode of the mth transverse order, respectively. {Q,,, , Qmn} are self coup- ling coefficients that represent the coupling among the fibre or slab modes due to the presence of the other. {K,,,, , K,,,,} are cross coupling coefficients, as defined by integrals 8 and 9 in Reference 9. These integrals were all solved [9] exactly except Qfoo, for which a large- argument asymptotic approximation of the modified Bessel function KO has to be used.

”

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Our closed-form expressions of the coupling coefli- cients are given below for the distributed coupling due to the curved fibre with a large curvature radius.

where A, = (n: - nf)/(n: - n:), B, = V,/V,, erf (x) is the error function and erfc (x) = 1 - erf (x). I, and I, are the modified Bessel functions of the first kind.

In the above expressions, the parameters A, and B, are introduced to quantify the geometric asymmetry of index profde of the slab waveguide. For the case involving a symmetric slab waveguide, we have A, = 0 and B, = 1. The parameter s represents the local distance between the surface of the fibre core and that of the overlay at any position z along the fibre length. The relationship between s, z and the curvature radius of the fibre R may be described by

Z2 s % s s , + -

2R

The numerical solutions of coupled-mode eqns. 4a and b and characteristics of the distributed coupling due to the curved fibre can be obtained by using a similar computer program (based on the Runge-Kutta method), as devel- oped in Reference 9 for the nondistributed coupling between a straight fibre and an asymmetric slab wave- guide, with the introduction of z-dependent s as given in eqn. 6 for all the coupling coetlicients. In the coupled- mode eqns. 4a and b, the allowable maximum order N , of the transverse slab modes with real propagation con- stants B, (m = 0, 1, 2, ..., N,) can be determined in the following way: from eqn. 2e, b, 3 0 or B. > U , , we have N,, = Int (B,D/n - 1/2) (Int: integer part of) while from eqn. 5, yc k on we have N,, = Int (y, D/n - 1/2). By definition, yf = js - nfk', i.e. y, < B,, so N, = Int (y, D / x - 1/2) is used in our calculation.

178

3 Numerical results and discussion

Numerical results are presented to mainly reveal the effect of the refractive-index asymmetry of the slab wave- guide on the distributed coupling, in addition to the effect of varying the fibre curvature. To allow comparison with the case of a straight fibre [ S I , all the optical and struc- tural parameters are as defined in Reference 9, except s, which is given by eqn. 6. The minimum distance between the fibre and the slab so = 0.5 pm and, if the fibre is straight ( R -+ a), s = so = 0.5 pm. All the numerical cal- culations, except where they are otherwise stated, have been carried out at a light wavelength 1 = 1.3 pm, with fibre radius a = 2.5 pm, slab thickness t = 3 pm, refractive index of the fibre cladding n, = 1.46, refractive index of the slab n, = 1.4745, D = 500pm and R = 50cm. It is also assumed that the light is launched initially through the fibre at z = zo < 0, i.e. ao(zo) = 1 and b,(z,) = 0, (n = 0, 1 , 2, ...), where the coupling is weak.

Note that the consistency and the convergence of our numerical results have been verified with respect to the value of D adopted above, which is typical within a range of valid values. For example, our numerical calculations have shown that the influence on the field distribution is negligible when D = 200 pm and indistinguishable when D 500 pm.

The propagation constant of the fibre LP,, mode BJo was obtained as the solution of the eigenvalue equation for weakly guiding fibres [lo], whilst the propagation constant B, of the slab mode was obtained using eqn. 2d, the eigenvalue equation for asymmetric slab waveguide and the propagation constants {B,} (n = 0, 1, 2, ...) of the discrete slab modes in transverse order were obtained using eqn. 2e. For simplicity, the slab mode of the ground order, i.e. m = 0 is used in the dispersion equation (eqn.

As used in the following numerical analysis, Table 1 shows the values of the fibre parameters, i.e. n J , V' and bfo whereas Table 2 shows those of parameters of the slab waveguide, i.e. no , V,, and 8,.

Table 1 : Parameters of the optical fibre, as used in Figs. 3 and 4

24.

Figure n, q=%) V,=&aJ(n:-n:) 8,. Effective flm-' index, ne,,

(a) 1.4817 0.015 3.101 7.1 247 1.4741 (b) 1.4756 0.01077 2.617 7.0987 1.4687

7.0942 1.4678 (c) 1.4745 0.01 2.520 (d) 1.4709 0.0075 2.177 7.0801 1.4649

Table2: Slab waveguide parameters, as used in each diagram of Figs. 3 and 4

C U N ~ n, A,c( = y) V,, = (W)J(n: - n:) 8,. pm-'

1.40 0.0492 3.3550 7.0932 !;; 1.46 0.0098 1.4955 7.1005 (3) 1.47 0.0030 0.8345 7.1 090

Calculated numerically from eqns. 4a and b, the scalar coupled-mode equations, Fig. 3a-d and Fig. 4a-d display the z-dependent absolute square mode-magnitude I a , l2 of the fibre when the fibre is straight ( R = CO) and curved (typically R = 50 cm), respectively. The above diagrams are numbered a d when the value of refractive index of the fibre core nJ is varied around that of the slab (n, = 1.4745), i.e. nJ > ns , nJ k n,, n f = n, and nJ < n,, respect-

IEE Proc.-Optoelectron., Vol. 142, No. 4, August 199s

ively, so that the effect of relative values of n, and n, can be shown in combination with that of the fibre curvature. Moreover, within each diagram of a-d the value of no is

also varied around that of n, (= 1.46), resulting in the three curves numbered 1, 2 and 3 with no < n, , no = n, and no > nc , respectively, so that the effect of relative values of no and n, is also shown.

1 .oo 0.98

0 96 0.98

a" 0.94 0 96

8 0 . 9 4

- -

N - 0 92 - 0.90

I

0 500 1000 1500 2000 2 . w 0

1.01, I

0 .90 0 ' 9 2 1 0 881 I I I

-3000 -Zoo0 -1000 0 loo0 2000 3000 z , Pm a

--.- I -3000 -2000 -1000 0 1000 2000 3000

z . um C

2 . w d

Fig. 3 Absolute square of the excitation coejjkients of thefibre mode, i.e. I a, l', as a function of the light propagation distance z along thefibre. Fibre is straight. a n, = 1.4817 b n, = 1.4756 c n, = 1.4745 d n, = 1.4709 For curves 1, 2 and 3, n, = 1.40, 1.46 and 1.47, respectively.

IEE Proc.-Optoelectron., Vol. 142, No. 4 , August 1995

01 I -3000 -Zoo0 -1000 0 1000 2000 3000

d z , w

Fig. 4 Absolute square of the excitation coeflcients of the fibre mode, i.e. I a, I', as a function of the light propagation distance z along the fibre. Curvature radius offibre R = 50 cm a d Refractive indices for fibre core, n,. and slab cladding, no, curves as s f i f i d in Fig. 3.

179

In Figs. 4 a 4 , the effect of the fibre curvature and the resultant distributed coupling are clearly observed, when compared with Figs. 3a-d. Apart from a dependence on the propagation distance, there appears to be an effective coupling region of the power-coupling during the light propagation, as a result of the varied local distance (s) between the fibre and slab waveguides. The effective coupling region may be defined by an onset point (z = z, < 0) as well as an offset point (z = z,,, r 0), beyond which the mode-power remains virtually unchanged because the coupling is too weak. This may be explained by the symmetric nature of the z-dependent parameter s, the local distance between the fibre and slab waveguides, which is related to the fibre curvature. In fact, all the coupling coefficients defined in 5&d are exponentially dependent on s as well as zz, as from eqn. 6, and this results in the distributed coupling strength.

Fig. 4a indicates that, when n, z n,, the launched power remains in the fibre initially until the light reaches a certain distance (zom rz -20oO pm) from the centre of the fibre half-block (z = 0 in Fig. 1). A very weak coup- ling appears at z = z,, with a little loss of the power as light propagates towards z = 0. The power is then gradu- ally recovered after the light goes through z = 0 and appears unchanged after a certain distance (zo,, z 2000pm). The curves of the power decay and recovery look symmetric about z = 0. This distributed coupling is in sharp contrast with Fig. 3a where the fibre is straight and the light power fluctuates between the fibre and the slab during the propagation but remains mainly in the fibre. It can also be observed that the greater the value of no, the weaker the coupling strength as the field of the slab modes is shifted further (along x direction in Fig. 1) from that of the fibre with an increasing value of no, the overlay refractive-index on the other side of the slab from the fibre.

As the value of n, gets close to (but still greater than) that of nsr Fig. 4b shows that when the fibre is curved, coupling is distributed and whether or not power is transferred from the fibre to the slab waveguide is strong- ly dependent on the value of n,. If n, < nc(A, < 0), the fibre power loss due to light propagation into the effect- ive coupling region is comparatively small and gradually recovered after the light goes through z = 0. If n, 2 nc(A, 2 0), the fibre power decay is so large when the light propagates into the effective coupling region that it cannot be recovered after the light goes through z = 0, that is, the light is largely coupled from the fibre to the slab waveguide. In Fig. 3b, when the fibre is straight, lowering the value n,, and that of Bl0, generally causes power oscillation or decay, depending on the values of no, and 8.. If no < nc(As < 0), the power oscillates with a high strength and dies out with only a fraction of the power coupled into the slab waveguide. If no 2 nc(A, 2 0), the fibre power also oscillates but with significant amount of power transferred to the slab.

When the value of n, equals that of nsr distributed coupling and interesting features of the fibre-mode power decay can be observed in Fig. 4c for the case of a curved fibre in comparison with the power beating and decay in Fig. 3c for a straight fibre. The lower the value of nor the higher the initial rate of the power decay. When the fibre is curved, if no < nc(AS < 0), the power oscillates at a high strength and the initial sharp power decay is partly com- pensated through coupling; if no a n,(A, 2 0), the power decay continues with much more of the power being transferred from the fibre to the slab. When the fibre is straight, the lower value of no and sharper attenuation

cause the power to oscillate at a higher strength and then die out with less power coupled from the fibre at the end.

When the value of n, is less than that of nsr Fig. 4d displays distributed coupling while both Fig. 3d and Fig. 4d show that the power is transferred in such a way that the lower the value of n o , the greater the part of power transferred, as expected due to the resultant stronger coupling; that is, more power goes to excite the contin- uum of modes in the slab waveguide.

The above observations on the distributed-coupling curves in Fig. 4a-d may be better appreciated in relation to the relative values of j?,o and pS, as shown in Table 1 and Table 2. When Bf0 > &, as for cases shown in Fig. 4a and curve 1 in Fig. 4b, the fibre mode is not in phase synchronism (i.e. not phase matched) with any of !he dis- crete slab modes introduced so a fraction of the power is coupled from the fibre to the slab waveguide and then recovered at the fibre output, suggesting that light, when guided in the slab, only propagates in the vicinity of the fibre. This can be verified by further studying the depend- ence of the total power at the centre of the slab on the transverse distance [6 , 71. When j?,, o Ps, as curve 2 in Fig. 4b and curve 1 in Fig. 4c illustrate, the fibre mode is close to phase synchronism with the slab mode and the power generally decays in the fibre and then oscillates with decreasing amplitude along with light propagation into the effective coupling region; if the value of p f 0 is slightly lower than that of &, the power is largely lost as radiation into the slab and the oscillation levels off when the coupling dies out gradually or, if the value of p f 0 is slightly higher than that of &, the power is transferred in part into the slab with a little recovery at the fibre output. Finally, when j?,o < j?,, curve (3) in Fig. 4b, curves 2 and 3 in Fig. 4c and curves 1-3 in Fig. 4d show a similar shape of power decay and the residual power in the fibre is dependent on the relative values of nf and n, as well as those of no and n,. For example, given the other optical and structural parameters of the fibre and the slab, increasing the allowable value of no (relative to those of n, and n,) should reduce the power loss to the slab waveguide.

Note that the effective coupling region mentioned above can be changed in size by varying the fibre radius of curvature or the minimum distance between the fibre and the slab. This is demonstrated in Figs. 5 and 6, respectively. In general, the effective coupling region is increased in size when the fibre radius of curvature is increased, or the distance between the fibre and the slab is reduced.

1 0 \<

0 2 t 0 1 I I I - 3 0 0 - 2 0 0 -loo0 0 100 2000 3000

2 . w

Fig. 5 All parameters as far curve 1 (Fig. 3 4 except R , which is varied as follows curve 1 R = 5 0 m curve2 R = 300cm curve 3 R = a

Effect of R, thefibre curvature on I a,, I* as a function ofz

I80 IEE Proc.-Optoelectron., Vol. 142, No. 4, August I995

The analysis so far has been applied to fibre-slab cou- plers of which the asymmetric slab waveguides support a single guided-mode. However, it should be straightfor- ward to extend our analysis to cases where the slab wave- guide may support two or more guidedmodes.

-Moo -2000 -1000 0 1000 2000 3000 2 . m

Fig. 6 as a Junction oJr All parameters as for curve 2 (Fig. 3b) except s, which IS varied as follows curve I s = 0.5 pm curve 2 s = 1.0 pm curve 3 s = 1.5 pm curve 4 s = 2.0 pm

Effect oJs, the distance between thefibre and the slab, on la, 1’

Fig. 7 is a typical dispersion diagram, showing dependence of the slab-mode effective indices labelled nef,(nefJ = jdk) on the slab thickness t. It can be seen that beyond certain cut-off thicknesses, higher-order (m > 0) slab modes may propagate in the slab waveguide.

0 5 10 15 20 25 30 35 40 t , wn

Fig. 7 nD = 1.40, nc = 1.46, n, = 1.4745, I = 3 pm Slab mode order, m, as follows curve I m = O curve 2 m = 1 curve 3 m = 3 curve4 m = 4

Dispersion diagram for slab waveguide

For a slab mode of mth order, varying the slab thickness generally causes a shift in the value of ne / / or b, and thus a shift of the fibre-slab coupling curves. This is illustrated in Fig. 8 when m = 0.

Assuming that only the LP,, mode is initially launched through the fibre and the propagation constant of the LP,, mode is close to that of the mth slab mode (determined readily from eqn. 24, the coupling between these two modes can be similarly analysed as above. This is because the fibre-slab mode coupling is strongly reson- ant, as shown here and previously [6, 8, 91, other slab

IEE Proc.-Optoelectron., Vol. 142, No. 4, August 1995

modes do not come into play as no substantial power will be built up in them through coupling.

z.Pm

Fig. 8 All parameters as for curve 2 (Fi& 36) except I which is varied as follows curve I t = 3.0 pm curve 2 I - 3.5 p m curve 3 I = 4.0 pm curve 4 t = 4.5 pm

Effect o f t , the slab thickness, on I a, 1’ as a function of L

4 Conclusions

An optical fibre half-block with a dielectric overlay has been studied theoretically. It was found that the charac- teristics of light coupling depend on the curvature of the fibre, the relative values of n/ and n, as well as those of n, and n, , that is, asymmetry of the slab waveguide index profile, as quantified by the parameter A,.

During the light propagation through the composite structure, the distributed coupling of light between the curved fibre and the slab waveguide is clearly demon- strated in comparison with the coupling between a straight fibre and a slab waveguide. The effective coup- ling region of the composite fibre-slab waveguide coupler can be varied by changing the curvature of the fibre or the minimum distance between the fibre and the slab waveguides.

When n, is lower than but not very close to n J , the light is well confined in the fibre core and is not substan- tially affected by the asymmetry due to the phase mis- match. However, as soon as n, is close to, equal or higher than n f (the two waveguides are close to phase synchronism), both the curvature of the fibre and the asymmetry of the slab cladding refective-index come strongly into play. The power decay and the amount of the power transferred from the fibre to the slab is quite different between the cases of a straight fibre and that of a curved fibre, and can be readily tuned by a small varia- tion in the value of no.

Apart from the variations of nf with respect to n, and no, to n, and nsr or vice versa, the features of the fibre- slab power coupling can also be readily tuned by varying the fibre curvature R, the slab thickness t and the minimum distance between the fibre and the slab.

This analysis and the features of the composite fibre- slab waveguide coupler composed of a fibre half-block with a slab overlay should find applications in many practical situations such as design of the coupler struc- ture and prediction of the coupling behaviours before fabrication, in relation to the fibre-slab materials, struc- tures and technical facilities available, thus enabling such couplers to achieve the power transfer performance desired by use of an optimal combination of the material, structural and fabrication parameters.

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2 JOHNSTONE, W., STEWART, G., CULSHAW, B., and HART, T.: ‘Fibre-optic polarisers and polarising couplers’, Electron. Lett., 1988.24, pp. 866-868

3 JOHNSTONE, W., MURRAY, S., THURSBY, G., GILL, M., McDONACH, A., MOODIE, D., and CULSHAW, B.: ‘Fibre optical modulators using active multimode waveguide overlays’, Electron. Lett., 1991,27, pp. 894-896

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7 ANDREEV, A.T., and PANAJOTOV, K.P.: ‘Distributed single- mode fibre to single-mode planar waveguide coupler’, J. Lightwave Technol., 1993,11, pp. 1985-1989

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9 ZHENG, S., BINH, L.N., and SIMON, G.P.: ‘Light coupling and DroDaaation in comwsite ootical fibre-slab wavermides’, J. Light- . . - waw Technol., 1995.i3, (2), pp. 244-251

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- 10 GLOGE, D.: ‘Weakly guiding fibres’, Appl. Opt., 1971, 10, pp. 2252-

182 I E E Proc.-Optoelectron., Vol. 142, No. 4, August 1995